11.02 Simplify ratios

Expressing ratios as fractions and percentages

If we wanted to describe the ratio of green dots to blue dots in the image above, we could write it as the ratio $2:3$2:3. Alternatively, if we wanted to express the ratio of green dots to the total amount of dots we could use the ratio $2:5$2:5.

We can also express ratios as fraction in these two ways. The ratio of green to blue dots is $\frac{2}{3}$23​, which we can interpret as there are two-thirds as many green dots as blue dots. We can also express the green dots as a ratio out of the total amount of dots with the ratio $\frac{2}{5}$25​, that is two-fifths of the total number of dots are green. When expressing ratios take care if you want a direct comparison between parts, or the proportion those parts are out of a whole.

 

Worked example

Example 1

A cordial mix requires the ratio of $30$30 ml of cordial to $100$100 ml of water.

(a) Express the ratio of cordial to water as a fraction in simplest form.

Think: As we want the ratio of cordial to water, the amount of cordial will be in the numerator and water will be in the denominator.

Do:

Ratio cordial to water	$=$=	$\frac{30}{100}$30100​
	$=$=	$\frac{30\div10}{100\div10}$30÷10100÷10​	Divide top and bottom by a common factor of $10$10 to simplify the fraction.
	$=$=	$\frac{3}{10}$310​

This means there is $\frac{3}{10}$310​ the amount of cordial compared to water. 

(b) Express the ratio of cordial to water as a percentage.

Think: From part (a) we know the fraction of cordial to water is $\frac{3}{10}$310​ . To convert a fraction to a percentage we multiply by $100%$100%. 

Do: $\frac{3}{10}\times100%=30%$310​×100%=30% 

(c) Write the ratio of cordial in a mixed drink as a fraction in simplest form.

Think: This time we want the ratio of cordial to the total liquid in a mixed drink. We have $30$30 ml of cordial and $100$100 ml of water, so the total will be $130$130 ml.

Do:

Ratio cordial to total	$=$=	$\frac{30}{130}$30130​
	$=$=	$\frac{30\div10}{130\div10}$30÷10130÷10​	Divide top and bottom by a common factor of $10$10 to simplify the fraction.
	$=$=	$\frac{3}{13}$313​

Thus, in a drink made to the given ratio cordial will be $\frac{3}{13}$313​ of the total drink.

(d) Express the ratio of cordial to the total drink as a percentage.

Think: From part (c) we know the fraction of cordial to water is $\frac{3}{13}$313​ . To convert a fraction to a percentage we multiply by $100%$100%. 

Do: $\frac{3}{13}\times100%=23.08%$313​×100%=23.08%

 

Practice questions

Question 1 

Question 2

Question 3

 

Equivalent and simplified ratios

Just as we can make equivalent fractions and can simplify fractions we can also do this with ratios written in the form $a:b$a:b. To make an equivalent fraction, recall we can multiply or divide the numerator and denominator by the same value. Having seen above how we can express a ratio as a fraction, this would be the same as multiplying or dividing both sides of a ratio by the same number. Multiplication and division by the same number preserves the proportion of the values in the ratio.

Consider the following example: a cake recipe that uses $1$1 cup of milk and $4$4 cups of flour, that is, the ratio of milk to flour is $1:4$1:4. What if we want to make two cakes? We would need to double the amount of milk and flour we use. This means we will need $2$2 cups of milk and $8$8 cups of flour. Now the ratio of milk to flour is $2:8$2:8.

But do we get two different ratios from the same recipe? No, the two ratios actually represent the same proportion of milk to flour. We say that $1:4$1:4 and $2:8$2:8 are equivalent ratios.

A ratio is a simplified ratio if there is no equivalent ratio with smaller integer values. This is the same as saying that the two integers in the ratio have a highest common factor of $1$1. Just as we simplify fractions by dividing by the numerator and denominator by a common factor we can divide both sides of a ratio expression by a common factor. 

A simplified ratio uses only integers. A ratio that includes fractions or decimals is not yet fully simplified and can be increased or decreased by an appropriate multiple to simplify it. 

 

Worked example

Example 2

A recipe for a salad dressing includes $20$20 ml of vinegar and $60$60 ml of olive oil.

(a) Fill in the table shown below to make equivalent ratios for a larger amount of dressing.

Vinegar	to	Olive Oil
$20$20	:	$60$60
$30$30	:	$90$90
$40$40	:	$\editable{}$
$\editable{}$	:	$300$300

Think: What do you need to multiply one side of the ratio by to get the known value? Multiply both sides by this value to get an equivalent ratio.

Do:

Vinegar	to	Olive oil		Vinegar	to	Olive oil
$20$20	:	$60$60		$20$20	:	$60$60
$\times2$×2		$\times2$×2		$\times5$×5		$\times5$×5
$40$40	:	$\editable{}$		$\editable{}$	:	$300$300

 

Hence, the completed table is:

Vinegar	to	Olive Oil
$20$20	:	$60$60
$30$30	:	$90$90
$40$40	:	$120$120
$100$100	:	$300$300

(b) What is the simplified ratio of vinegar to olive oil in the dressing?

Think: What is the highest common factor of $20$20 and $60$60? Divide both sides of the ratio by this number.

Do:

Vinegar	to	Olive oil
$20$20	:	$60$60
$\div20$÷20		$\div20$÷20
$1$1	:	$3$3

 

The simplified ratio of vinegar to olive oil is $1:3$1:3. A simplified ratio is great for simple recipes, this one tells us we need three times as much olive oil than vinegar. So if rather than measuring in millilitres we put $1$1 tablespoon on vinegar we know we need to put $3$3 tablespoons of olive oil.

 

Practice questions

Question 4

Question 5

Question 6

Question 7